Ghost images and feasibility of reconstructions with the Richardson-Lucy algorithm

摘要

This paper is the result of a question that was raised at the recent workshop on ''The Restoration of HST Images and Spectra H'', that took place at the Space Telescope Science Institute in November 1993, for which there was no forthcoming answer at that time. The question was: What is the null space (ghost images) of the Richardson-Lucy (RL) algorithm.. Another question that came up for which there is a straight-forward answer was: What does the MLE algorithm really do.. In this paper we attempt to answer both questions. This paper will begin with a brief description of the null space of an imaging system, with particular emphasis on the Hubble telescope. The imaging conditions under which there is a possibly damaging null space will be described in terms of linear methods of reconstruction. For the uncorrected Hubble telescope, it is shown that for a PSF computed by TIN on a 512 (times) 512 dimension, there is no null space. We introduce the concept of a ''nearly null'' space, with an unsharp distinction between the ''measurement'' and the ''null'' components of an image and generate a reduced resolution Hubble Point Spread Function (PSF) that has that nearly null space. We then study the propagation characteristics of null images in the Maximum Likelihood Estimator (MLE), or Richardson-Lucy algorithm, and the nature of its possible effects, but we find in computer simulations that the algorithm is very robust to those effects: if they exist,the effects are local and tend to disappear with increasing iteration numbers. We then demonstrate how a PSF that has small components in frequency domain results in noise magnification, just as one would expect in linear reconstruction. The answer to the second question is given in terms of the residuals of a reconstruction and the concept of feasibility.